The Continuum Hypothesis, Part I

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Author: W. Hugh Woodin

Publication: June/July 2001

Journal: Notices of the AMS. Volume 48, number 6.

Introduction

Arguably the most famous formally unsolvable problem of mathematics is Hilbert's first problem:

Cantor's Continuum Hypothesis: Suppose that X \subseteq \mathbb{R} is an uncountable set. Then there exists a bijection \pi : X \rightarrow \mathbb{R} .

This problem belongs to an ever-increasing list of problems known to be unsolvable from the (usual) axioms of set theory.

However, some of these problems have now been solved. But what does this actually mean? Could the Continuum Hypothesis be similarly solved? These questions are the subject of this article, and the discussion will involve ingredients from many of the current areas of set theoretical investigation. Most notably, both Large Cardinal Axioms and Determinacy Axioms play central roles.

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